Exercise 4.4 — Angle Sum of Triangle

Angle Sum Property of a Triangle

Two of the most important theorems in Class 9 Geometry come together in Exercise 4.4 of Chapter 4 — Lines and Angles. The first is the Angle Sum Property: the three interior angles of any triangle always add up to exactly 180°. The second is the Exterior Angle Theorem: when a side of a triangle is extended beyond a vertex, the exterior angle formed equals the sum of the two non-adjacent interior angles (called the remote interior angles). These two results are tested heavily in CBSE, Telangana, and Andhra Pradesh board exams, and every question in this exercise uses one or both.

Understanding the Exterior Angle Theorem

When side AC of triangle ABC is extended to point D, the angle ∠BCD formed outside the triangle is the exterior angle at C. The theorem tells us that ∠BCD = ∠A + ∠B. This is not just a formula to memorise — it follows directly from the angle sum property combined with the linear pair property. Understanding this connection is what allows you to solve multi-step problems confidently.

• Direct application (Q1, Q6, Q8) — When two interior angles are known, the exterior angle is simply their sum. For example, if ∠A = 50° and ∠B = 60°, the exterior angle at C is 110°.
• Working backwards (Q8) — When the exterior angle and one interior angle are known, subtract to find the third. In Q8, ∠SPR = 135° and ∠PQR = 70°, so ∠PRQ = 135° − 70° = 65°.
• Chaining two exterior angles (Q11, Q17) — The exterior angle of one triangle becomes an interior angle of another. Apply the theorem step by step to each triangle separately.
• Ratio problems (Q14) — If the exterior angle and the ratio of two interior angles are both given, the two interior angles share the exterior angle value in that ratio. Divide the exterior angle according to the ratio to find each angle.

Worked Example – Angle Bisectors Inside a Triangle (Q9)

In triangle XYZ, ∠X = 62° and ∠XYZ = 54°. YO and ZO are the bisectors of ∠XYZ and ∠XZY respectively, meeting at O. To find ∠OZY and ∠YOZ, first find ∠XZY using the angle sum property: ∠XZY = 180° − 62° − 54° = 64°. Since ZO bisects ∠XZY, ∠OZY = 64° ÷ 2 = 32°. Since YO bisects ∠XYZ, ∠OYZ = 54° ÷ 2 = 27°. Finally, applying the angle sum property to triangle OYZ gives ∠YOZ = 180° − 32° − 27° = 121°. Notice that the angle at O is always obtuse — this is a useful self-check in any bisector problem.

Worked Example – Combining Parallel Lines with Triangles (Q3)

When parallel lines appear alongside triangles, you use both sets of tools in the same problem. In Q3, AB ∥ CD and BC ∥ DE. Since AB ∥ CD, the angle 3x° and the 105° angle are alternate interior angles, giving 3x = 105°, so x = 35°. Since BC ∥ DE, angle D also equals 105° (alternate interior angles). Then applying the angle sum property to triangle DCE: 105° + 24° + y = 180°, giving y = 51°. The key habit is to identify the parallel pair first, find the angles it controls, and only then apply the triangle property.

Common Mistakes to Avoid

• Confusing interior and exterior angles — The exterior angle equals the sum of the two non-adjacent interior angles, not the adjacent one. The adjacent interior angle and the exterior angle form a linear pair (they add to 180°), which is a different relationship entirely.
• Forgetting to find the interior angle first — In Q8, ∠PQT = 110° is given but this is outside the triangle. Always find the interior angle first using the linear pair: ∠PQR = 180° − 110° = 70°, then apply the exterior angle theorem.
• Bisector problems: halving the wrong angle — In Q9, students sometimes halve ∠X instead of ∠Y and ∠Z. The bisectors are YO (bisecting ∠Y) and ZO (bisecting ∠Z), so only those two angles are halved.
• Isosceles triangles: assuming all angles are equal — In Q16, AB = AC means ∠ABC = ∠ACB, not that all three angles are equal. Use the angle sum property with the two equal angles to find each one correctly.
• Missing the chain step in Q11 and Q17 — When two triangles share a vertex or an angle, the exterior angle of the inner triangle is an interior angle of the outer one. Solve the inner triangle first, then use that result in the outer triangle.

What This Exercise Prepares You For

The angle sum and exterior angle theorems are the foundation of triangle geometry throughout Class 9 and Class 10. They are used directly in the Triangles chapter to prove congruence and similarity conditions. The bisector result from Q9 reappears in the incenter topic. The method of combining parallel lines with triangle angle sums (Q3, Q10, Q15) is tested in Class 10 similar triangles and in coordinate geometry proofs. For students following the Exercise 4.3 path, this exercise is the natural next step, moving from straight lines to closed figures.